A number is a multiple of 3 if you can add up the digits and the sum is also a multiple of 3. But why does it work that way? And do similar phenomena appear in other base number systems?

Question

A number is a multiple of 3 if you can add up the digits and the sum is also a multiple of 3. But why does it work that way? And do similar phenomena appear in other base number systems?

In: 17

Answer 1

Because if you have a number xyz it can also be written as 100x + 10y + z. The 100 and 10 multiplicators can be rewritten as 9+1 or 99+1. You can discard the 9 and 99 because they are multiples of 3 (and 9 that’s why it also works for 9) so xyz=100x + 10y + z = x+y+z +(99x + 9y)